Filtration (probability theory)

In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are totally ordered collections of subsets that are used to model the information that is available at a given point and therefore play an important role in the formalization of random (stochastic) processes.

Definition

Let $(\Omega ,{\mathcal {A}},P)$ be a probability space and let $I$ be an index set with a total order $\leq$ (often $\mathbb {N}$, $\mathbb {R} ^{+}$, or a subset of $\mathbb {R} ^{+}$).

For every $i\in I$ let ${\mathcal {F}}_{i}$ be a sub-σ-algebra of ${\mathcal {A}}$. Then

$\mathbb {F} :=({\mathcal {F}}_{i})_{i\in I}$

is called a filtration, if ${\mathcal {F}}_{k}\subseteq {\mathcal {F}}_{\ell }$ for all $k\leq \ell$. So filtrations are families of σ-algebras that are ordered non-decreasingly. If $\mathbb {F}$ is a filtration, then $(\Omega ,{\mathcal {A}},\mathbb {F} ,P)$ is called a filtered probability space.

Example

Let $(X_{n})_{n\in \mathbb {N} }$ be a stochastic process on the probability space $(\Omega ,{\mathcal {A}},P)$. Let $\sigma (X_{k}\mid k\leq n)$ denote the σ-algebra generated by the random variables $X_{1},X_{2},\dots ,X_{n}$. Then

${\mathcal {F}}_{n}:=\sigma (X_{k}\mid k\leq n)$

is a σ-algebra and $\mathbb {F} =({\mathcal {F}}_{n})_{n\in \mathbb {N} }$ is a filtration.

$\mathbb {F}$ really is a filtration, since by definition all ${\mathcal {F}}_{n}$ are σ-algebras and

$\sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).$

This is known as the natural filtration of ${\mathcal {A}}$ with respect to $(X_{n})_{n\in \mathbb {N} }$.

Types of filtrations

Right-continuous filtration

If $\mathbb {F} =({\mathcal {F}}_{i})_{i\in I}$ is a filtration, then the corresponding right-continuous filtration is defined as

$\mathbb {F} ^{+}:=({\mathcal {F}}_{i}^{+})_{i\in I},$

with

${\mathcal {F}}_{i}^{+}:=\bigcap _{i<z}{\mathcal {F}}_{z}.$

The filtration $\mathbb {F}$ itself is called right-continuous if $\mathbb {F} ^{+}=\mathbb {F}$.

Complete filtration

Let $(\Omega ,{\mathcal {F}},P)$ be a probability space, and let

${\mathcal {N}}_{P}:=\{A\subseteq \Omega \mid A\subseteq B{\text{ for some }}B\in {\mathcal {F}}{\text{ with }}P(B)=0\}$

be the set of all sets that are contained within a $P$-null set.

A filtration $\mathbb {F} =({\mathcal {F}}_{i})_{i\in I}$ is called a complete filtration, if every ${\mathcal {F}}_{i}$ contains ${\mathcal {N}}_{P}$. This implies $(\Omega ,{\mathcal {F}}_{i},P)$ is a complete measure space for every $i\in I.$ (The converse is not necessarily true.)

Augmented filtration

A filtration is called an augmented filtration if it is complete and right continuous. For every filtration $\mathbb {F}$ there exists a smallest augmented filtration ${\tilde {\mathbb {F} }}$ refining $\mathbb {F}$.

If a filtration is an augmented filtration, it is said to satisfy the usual hypotheses or the usual conditions.

See also

• Natural filtration
• Filtration (mathematics)
• Filter (mathematics)